Mahler's conjecture and duality in convex geometry

Original Announcement

This workshop will be devoted to
duality problems in convex geometry,
which deal with relations between convex bodies and their polar
bodies. The participants will explore
the opportunities opened by a flurry of recent results related to the
problem, most of which are based
on new promising analytic techniques.

The main topics for the workshop are:

The volume product of a convex body $K$ in $\R^n$ is defined by
$P(K)=vol_n(K)vol_n(K^*)$ where $K^*$ is the polar body of $K$.
Mahler's conjecture asks whether
the minimum of the volume product in the class of origin-symmetric convex
bodies is attained at the unit cube. Despite many important partial results,
the problem is still open in dimensions 3 and higher. The participants will
explore the opportunities opened by new proofs of the Bourgain-Milman theorem
(establishing Mahler's conjecture up to an absolute constant).

It has been known for a long time that many results on sections and
projections
of convex bodies are dual to each other, in the sense that sections
of a body behave
in a similar way to projections of the polar body. Methods of Fourier
analysis can be
applied to develop a unified approach to some of these results. The
participants will try
to extend this Fourier approach to other problems, in particular to
the question of whether
intersection and polar projection bodies are isomorphically equivalent.